Yield Analysis and Performance Optimization Using FastBlaze and
SPAYN

Conventional device simulators often suffer from slow
execution times, leading to a trade off between mesh density and physical
model complexity against CPU run time and convergence. This requires engineers
to compromise accuracy to achieve a reasonable throughput.

By focusing solely upon FET simulation, FastBlaze
has been able to greatly optimize the physical solution procedure, enabling
the use of the most sophisticated physical models while maintaining a
fast execution (typically less than 1 minute for a full set of DC ID-VDS
curves) This speed enables both complex and extensive experimental designs
to be completed on a reasonable time scale, permitting experiments that
would be prohibitively expensive for more traditional device simulators.

There are two common distribution choices for Monte Carlo
experimental design, a uniform random distribution suits device optimization,
as the input parameter plane is more efficiently covered, however, by
using a Gaussian profile, which is more representative of natural variations
in the device, we can also perform yield analysis directly upon the experimental
output.

Device Optimization

When the results from the Monte Carlo experiment are
investigated in SPAYN the user can identify "good"
devices by visually inspecting the scatter plots. In Figure 1 a device
has been chosen with both high gmpk and breakdown. SPAYN
then displays each of the input variables for that simulation in a pop-up
box.

A simple yet effective method of optimization is then
to center the input values on this "good" device and re-run
the whole experiment. Figure 3 displays the combined scatter plots for
both experiments, showing a marked improvement in both gmpk
and VDbrk.

Figure 4 shows histograms of both gmpk
and VDbrk for the nominal and optimized cases. The
qualitative improvement in the parameters is immediately observable, with
the mean gmpk increasing from 410 to 490 mS/mm and
VDbrk from 24 to 28 V. Further there is no significant
change in the standard deviation of either distribution.

Figure 4. Distribution analysis. Here
the mean value of both VDbrk and gmpk have been increased in the
optimized structure with no significant change in the standard
deviations.

Yield Analysis

Since both experiments were performed using Gaussian
distributions the output distribution functions can be evaluated directly
to obtain the yield values.

In our example we choose an arbitrary "fail"
point for the breakdown voltage and integrate both the nominal and optimized
distributions to calculate the expected yield from a wafer. Choosing a
minimum of 20 Volts breakdown the yield from the nominal structure is
92.2% where as the optimized is greater than 99%. If the minimum is shifted
to 22 Volts, the nominal structure's yield drops to 80% where as the optimized
is still greater than 98%.

SPAYN Parameter Analysis

The previous examples demonstrate a straight forward
optimization technique, however, the inter-dependencies of each input
parameter have not been analyzed. SPAYN provides the tools
for investigating these relationships via the correlation matrix.

SPAYN can also be used to generate an analytical
"black box" model of this data set through regression analysis.
This is more computationally efficient than FastBlaze however
is strictly limited to this structure.

For more information on the techniques described below
please refer to the SPAYN User's Manual.

Correlation

SPAYN can be used to investigate the inter-parameter
dependencies through the correlation matrix.An abridged matrix showing
the most significant variables is presented in Table 2.

This is useful when determining which parameters will
have a larger influence in the regression models. In this case we can
see that the gate length, second recess length and the total recess depth
are highly correlated with VDbrk (|r| > 0.5).

Regression

Regression analysis can be performed within SPAYN
to generate response surfaces for the target parameters. These regression
equations can then be used to predict new values for the response variables
far more efficiently than by using any physical simulator.

Using VDbrk as the response variable
and the 9 predictor parameters from Table 1, an analysis of variance (ANOVA)
was performed to identify the most suitable model. If a more complete
ANOVA analysis was required, an engineer might also add or remove individual
parameters.

There are several model selection criterion available,
the most commonly used being p-value and adjusted R2.

From the ANOVA table (table 3) all of the p-values up
to model 3. are highly significant, hence our regression model should
include all linear (x), interaction (xy) and quadratic terms (x).
Adding extra parameters to the regression equation (models 4,5 and 6)
produce non-significant p-values, indicating that it is not worth including
them in the model.

Table 3. Analysis of variance (ANOVA) for VDbrk.

As an alternative we can also use the adjusted R
value as the selection criterion. In this case model 4 should be used
i.e. all linear (x), interaction (xy), quadratic (x2) and interaction
(xyz) terms.

The computational demands for evaluating a regression
model are minimal when compared to a full physical simulation, hence we
chose model 4 from the ANOVA table.

After fitting the regression equation a residual analysis
is performed to check the model. This is accomplished by plotting the
residuals against the estimated values and also individual predictor variables
(see Figure 5)

Figure 5. Residual analysis of model 4.

A visual inspection of these plots should not reveal
any discernible trends. If any patterns were apparent this would indicate
a poor model and further analysis would be required.

Finally the response surface was then used in place of
FastBlaze to re-generate the original Monte Carlo data.
Note: 1000 samples were simulated in under 1 second on a Sun ULTRA 10
workstation. VDbrk was then compared between the two data sets, the first
generated by FastBlaze and the second via the SPAYN
regression model. A deviation of less than 5% was typically observed,
illustrating the validity of this approach.

Conclusions

By looking at the input parameters from the simulated
"good" devices we can draw some conclusions about general design
criteria for this type of device. First we have obvious changes, decreasing
the gate length and increasing the second recess length to increase gm
and breakdown respectively. The recess fraction, the ratio of 1st and
total recess depths should be fixed for this structure at 0.8. Finally,
the doping density of the 1st delta should be lowered to increase breakdown,
whilst raising the density of the 2nd delta to maintain gm.

The SPAYN statistical analysis was used
to confirm the device optimization and corresponding improvement in yield.
Further, the correlation matrix revealed which input parameters are most
significant.

An analysis of variance was performed to select an appropriate
regression model which was checked with a residual analysis. Figure 5
shows no discernible trends indicating a good model. Finally the regression
equation was used to predict new values for the response variables with
good accuracy validating the model selection.